Which of the following functions are decreasing on (0, π/2)?
Explanation
Option (A) is correct because Example 9 in the context explicitly states that the function f(x) = cos x is decreasing in (0, π). Since (0, π/2) is a subset of (0, π), cos x is decreasing on (0, π/2). For the other options: (B) cos 2x has derivative -2 sin 2x, which is negative on (0, π/2), so it's also decreasing, but this isn't explicitly stated in the context. (C) cos 3x is not decreasing on the entire interval (0, π/2) because its derivative changes sign. (D) tan x has derivative sec²x > 0, so it's increasing, not decreasing.
Solution Steps
Step 1: From Example 9, cos x is decreasing in (0, π). Since (0, π/2) ⊂ (0, π), cos x is decreasing on (0, π/2).
Step 2: For cos 2x, f'(x) = -2 sin 2x < 0 on (0, π/2), so it's also decreasing.
Step 3: For cos 3x, f'(x) = -3 sin 3x changes sign on (0, π/2), so it's not decreasing on the entire interval.
Step 4: For tan x, f'(x) = sec²x > 0, so it's increasing.
Step 5: Since (A) is directly supported by Example 9, the answer is (A).